POSSIBILITY OF THERMAL INSTABILITY FOR 4-VANE RFQOPERATION WITH HIGH HEAT LOADING

V.V. Paramonov∗

Institute for Nuclear Research of the RAS, Moscow, Russia

Abstract

Due to dispersion properties, 4-vane RFQ cavity withoutresonant coupling is a thermally unstable structure. Withdeterioration of balance for local detuning there is a possi-bility for runaway in the field distribution and related ther-mal effects. It can results, in principle, in irreversible plas-tic deformations and cavity frequency shift. Both the in-crement and the threshold of instability are proportional tothe average dissipated RF power. This possibility is moreprobable for long RFQ cavities. Also particularities for thecavity ends design are important. Some general features ofthis effect are discussed and illustrated with simulations.

INTRODUCTION

The Normal Conducting (NC) RFQ cavity is now theimprescriptibly part of the hadron’s linac. For proton linac4-vane RFQ with operating TE210 mode is now prevailing.The sensitivity of the field longitudinal distribution to theshape deviations in such cavities strongly depends on therelative RFQ length Lc

λ0, where Lc is the RFQ length and

λ0 is the operating wavelength. To reduce it for long RFQcavities, Lc

λ0≥ 5 the resonant coupling with special cou-

pling cells was proposed [1], providing for RFQ propertiesof compensated structure in the field distribution sensitivityand stability. The shorter RFQ cavities, Lc

λ0≤ 5, as a rule,

are without coupling cells. It this report the stability of thelongitudinal field distribution in time for RFQ without res-onant coupling is considered with respect thermal inducedgeometry perturbations.

OPERATING REGIME STABILITY

In modern proton linac’s RFQ operate with the fre-quency f0 ∼ (324 ÷ 402.5)MHz, maximal electric sur-face field Esm ∼ 1.8Ek ≈ (25 ÷ 32)MV

m , which corre-sponds to the maximal magnetic field at the regular RFQsurface Hsm ∼ 5.2kA

m . It results in the pulse heat dis-sipation is of Pp ≈ 100kW

m . Even for operation withduty factor df ∼ (1 ÷ 6)% the average heat dissipationPa ∼ (1 ÷ 6)kW

m is significant for thermal effects. Thetemperature of the cavity increases and f0 decreases due tothe cavity expansion. For the fixed cooling conditions thecavity frequency shift df0 is linearly proportional to Pa.Suppose a steady-state high RF power operation with a ref-erence field distribution in the cavity is achieved. Let ussuppose a small local temperature deviation δT > 0 at

∗ paramono@inr.ru

the cavity surface due to some reasons. It may be eithercooling fluctuation or electric discharge. This local tem-perature deviation leads to the local cavity expansion andlocal frequency change δf . The local frequency change δfimmediately results in the change of the field distributionalong the cavity and the change of the field in place of lo-cal heating. Depending on the cavity dispersion properties,two options, shown in Fig. 1, are possible.In the first case, Fig. 1a, the local field relatively decreases.

Figure 1: Thermal stable (a) and unstable (b) chain.

The local RF power dissipation decreases, the local temper-ature decreases, the local frequency increases, canceling orreducing initial frequency deviation δf . It is the stable case.After some time the cavity returns to operation with the ref-erence field distribution.In the second case, Fig. 1b, the local field, togetherwith the local RF power dissipation, relatively increases,the local temperature increases, the local frequency de-creases, amplifying initial local frequency deviation δf .Self-amplifying runaway starts and in the cavity itself thereis no physical mechanism, which can stop it.The thermal stability of operation for multi cells NC cavi-ties was considered in [2]. All time the compensated struc-tures can be tuned for stability conditions by appropriatechoice of the stop-band width, [3], [2]. The simple multicells structure with operating 0-mode and positive disper-sion is unstable. RFQ cavity has no marked cells, but op-erating mode TE210 is zero type mode and dispersion forTE21n modes is positive.The deviation of the quadruple field longitudinal distribu-tion along RFQ is the result of TE21n modes superposition.The field distribution �E in the cavity with a small dimen-sion deviation ΔV can be described as [4]:

�E = �E0 +∑

n

�Enf2n

∫ΔV

(Z20

�H0�H∗

n − �E0�E∗

n)dV

W0(f20 − f2

n), (1)

where Z0 =√

μ0ε0

,∫

V Z20

�Hn�H∗

mdV =∫

V�En

�E∗mdV =

δnmW0 and V is the cavity volume, �E0, �H0 are the electricand the magnetic field distributions for operating TE210

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2C RFQs 545

mode and fn, �En, �Hn are the frequency and fields distribu-tions for TE21n mode. The similar relation for descriptionof a field perturbation in the chain of coupled cavities isgiven in [5] with a matrix form.In 4-vane RFQ without coupling cells the frequency andfield distribution along the cavity for TE21n mode are:

f2n � f2

0 + (nc

2Lc)2, �En �

√2 �E0cos(

nπz

Lc), (2)

where c is the velocity of light. Transforming (1) with sug-gestion the volume deviation ΔV is placed at z = z0, tak-ing into account (2) and using frequency perturbation the-orem, one can get:

�E � �E0 −∑

n

√8 �E0

δf0

f0(1 +

4L2c

n2λ20

)cos(nπz0

Lc). (3)

As one can see from (3), the negative local frequency de-viation δf0 < 0 at the cavity end leads to the local fieldincreasing δ| �E| = | �E − �E0| > 0. Without resonant cou-pling 4-vane RFQ cavity is thermal unstable.Most dangerous is the detuning of the cavity ends, z =0, Lc. This effect linearly rises with the cavity length Lc

λ0,

because the relative detuning δf0f0

, caused the same abso-lute local deviation δf0 is inverse proportional to Lc. The

dependence of δ| �E||�E0|

increasing on the cavity length Lc

λ0is

shown in Fig. 2, assuming the detuning δf0 = −10−3f0 ofthe cavity part with 0.1λ0 at the cavity end z = 0.

Figure 2: The longitudinal field distribution sensitivity tothe RFQ end detuning δf .

RFQ ENDS DETUNING

The RFQ vanes have undercuts in the cavity ends to re-turn magnetic field flux, to tune the cavity frequency andtune longitudinally the field distribution. Different shapesare known - the undercut with a small tip (N1 in Fig. 3a,[6]), the undercut with a moderate tip (N2 in Fig. 3b, [7])and mostly distributed undercut with inclined tip, see, forexample SNS RFQ, [8]. The last one can be realized bothwith vertical (N3 in Fig. 3c) and inclined (N4 in Fig. 3d)outputs of the vane cooling channel.

For all undercut options the maximal value of magneticfield and related dissipated heat density takes place at thecavity ends. Additionally, the vane cooling channel may beat enlarged distance from the undercut tip, as one can seefrom Fig. 3. All options, shown in Fig. 3, were tuned for

Figure 3: Different options for vane undercuts and coolingchannels. 1 - vane undercuts, 2 - the channels for cavitybody cooling, 3 - the vane channel.

operating frequency and a flat field distribution along thecavity. Initial thermal-stress analysis has been performedaccording [9] in engineering approach assuming the dutyfactor of 3% (Pa = 0.03Pp) and the average flow velocityVav = 1.5 m

sec . The temperature distributions are shown inFig. 4.

The undercuts with small and moderate tips (N1 and

Figure 4: The surface temperature distributions and max-imal temperature rise dT values at the vane undercuts fordifferent cavity end design.

N2 in Fig. 3a,b) have the larger surface temperature risedT = Tmax − Tw with respect to the temperature of cool-ing liquid. The smallest dT value has the undercut withinclined tip and inclined channel output, N4 in Fig. 3d.The distributions of the thermal induced displacements andrelated frequency shift δf0 values are shown in Fig. 4. Forδf0 value definition it is assumed, that the detuning is lo-calized near cavity end at the length 0.1λ0.

Even from the qualitative displacement distributions inFig. 5 one can see the significant difference between op-tions N1, N2 and N4 - small and moderate tips move tothe cavity end plate. For the inclined undercut with in-clined channel the plates moves from the tip, decreasingcapacitance. More detailed analysis shows for all undercutboth longitudinal and radial (dr < 0) displacements. Theminimal δf0 value provides N4 option due to better cool-ing and smaller induced displacements.The direction of cooling water is very important and can beeven decisive. In more details it is presented in [10].Depending on the design and cooling particularities, ends

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Figure 5: The displacements distributions and induced fre-quency shift values for different undercuts options.

of RFQ cavity have different values for ’power sensitivity’∂f0∂Pa

, which is not the same as for regular RFQ part.

INCREMENT AND THRESHOLD

The instability is the result of the coupled RF - thermal- stress - RF interaction and the increment value ζ dependson both cavity RF parameters and cavity material thermaland mechanical properties, details of the design.Assuming the instability development as δf0 ∼ A0e

ζt weget dδf0

dt = ζδf0. Let us consider the sequence d(δf0) ∼δx ∼ δT ∼ δP ∼ δE ∼ δf0. From (3) it follows

δP ∼ PaL2

c

λ20

δf0f0

. From the equation for thermo-elastic de-

formations follows δx ∼ αEc

δT , where α and Ec are thelinear expansion coefficient and Young module for copper.The time scale of the instability is defined by the slowestprocess in the sequence - heat propagation, which qualita-tively describes by thermal diffusivity Dc = Kc

ρcCc, where

Kc, ρc, Cc are the heat conductivity, density and specificheat values for copper. The dimensions of cavity parts xi

are also important, [10]. Because instability is related tolocal effects, it depends on the local RF power loss densityP s

a ∼ Pa

x2i

. But instability location is not known directly

and outside the cavity we see the total average power Pa.For this reason we use more usual Pa term in ζ value es-timation. The time scale of instability is shorter, than thetime constant of the cavity for RF power input, [10]. Sum-marizing, we can write expression for instability incrementas:

δf0 ∼ A0eζt, ζ � B0

Pa

x2i

L2c

λ0cV

αDc

Ec, A0, B0 = const

(4)In the frame of presented model, instability starts whenthere is a powerful RF supply to support essential thermaleffects, and one part of the cavity has the negative local fre-quency detuning with respect total cavity. Most likely can-didates are cavity ends, due to stronger effect on field dis-tribution higher in (3) and essentially different ∂f0

∂Pavalue.

The cavity tuning for frequency and field distribution isat RF power signal level, when thermal effects are absent.The RFQ cavity input and output ends, as a rule, have dif-

ferent design and may have different fields values. If aftertuning the ends are not balanced in frequency detuning andone end has initial δf0 ≤ 0, will be startup instability si-multaneously with RF power switch on.For balanced ends and correct cooling scheme there is apower range 0 ≤ Pa ≤ P cr

a for RFQ stable operation,when the balance of local detuning is preserved. There area lot of operating 4-vane RFQ’s, but not with very highaverage RF power. But, due to different ∂f0

∂Pavalues, with

Pa increasing this balance can become weaker and at somevalue Pa = P cr

a , which depends on cavity design, cooling,tuning and control, will be violated - one cavity end willget negative detuning. It will be start of instability.In frame of presented model, the instability threshold de-pends on the average power, dissipated in the cavity. Ad-ditional heat sources at the ’weak’ cavity part, like electricbreakdowns and particle losses [8], can provoke instabilityor decrease threshold slightly. In more details instabilitythreshold is discussed in [10].

SUMMARY

Consideration shows, that due to dispersion properties,4-vane RFQ cavity without resonant coupling has the prop-erty of thermal instability. If instability started, the cavityhas no own mechanism, except inelastic (irreversible) de-formation to stop it. Both the increment and the thresholdof instability depend on the average RF power, dissipated inthe cavity and can limit possible duty factor value for eachparticular design. Control system for longitudinal field dis-tribution together with fast movable tuners can dump insta-bility.

REFERENCES

[1] M.J. Browman, L.M. Young. Coupled RFQ’s as Compen-sated structures, Proc. 1990 Linac, p. 70, 1990

[2] V. Paramonov, Stability of Normal Conducting StructuresOperation ... Proc. 2008 Linac Conf., p.61, (2008)

[3] L.M. Young, J. M. Potter. CW Side Coupled Linac for theRacetrack Microtron. Proc. PAC 1981, p. 3508, (1983)

[4] B.P. Murin (ed), Ion Linear Accelerators, v.2, Atomizdat,Moscow, 1978, (in Russian)

[5] E.A. Knapp et al., Coupled cavity model for standing waveaccelerator tanks. Rev. Sci. Instr., v. 38, p. 1583, (1967)

[6] A. Ueno, Y. Kondo, RF-Test of a 324-MHz, 3-MeV, H- RFQStabilized with PISL’s, Proc. 2002 Linac, p. 545, (2002)

[7] G. Romanov et al., The Fabrication and Initial Testing of theHINS RFQ. Proc. 2008 Linac, p. 160, (2009)

[8] S. Kim et al., Stabilized operation of the SNS RFQ, Phys.Rev. STAB, v. 13, 070101, 2010

[9] A. Skassyrskaya et al., The complete 3-D coupled rf-thermal-structural-rf analysis ... Proc. 2002 Linac, p. 216,(2002)

[10] V. Paramonov, One Aspect of Thermal Stability for 4-vaneRFQ Operation. Proc. 2010 RuPAC, THPSC014, Protvino,28 Sep., 1 Oct. (2010)

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